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Kleismic family: Difference between revisions

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m →2.3.5.7.13 subgroup: S26*S27 (ratwolfsma) and S25/S27 (ragisma) omitted due to being trivially implied; note that there are other trivially implied commas listed here only because they have nontrivial equivalences
Rework S-expressions as part of the description
Line 23: Line 23:
* 5-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 5-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)
* 5-odd-limit diamond monotone and tradeoff: ~6/5 = [315.641, 317.263]


{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 882c }}
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 882c }}
Line 34: Line 33:


; Analysis and diagrams:
; Analysis and diagrams:
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by [[David Keenan]]


=== Cata ===
=== Cata ===
Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The S-expression-based comma list of the temperament is {[[325/324|S10/S12 = S25*S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]}.
{{Main| Catakleismic }}
{{Main| Catakleismic }}


Line 42: Line 43:


Comma list: 325/324, 625/624
Comma list: 325/324, 625/624
S-expression-based comma list: {[[325/324|S10/S12 = S25*S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]}


Sval mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
Sval mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
Line 70: Line 69:
* 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]
* 7-odd-limit diamond monotone and tradeoff: ~6/5 = [308.744, 322.942]
* 9-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]


{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
Line 89: Line 86:
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 11-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]


{{Optimal ET sequence|legend=1| 4, 15, 19, 34 }}
{{Optimal ET sequence|legend=1| 4, 15, 19, 34 }}
Line 107: Line 103:
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~6/5 = 315.789


{{Optimal ET sequence|legend=1| 4, 15f, 19, 53def, 72def }}
{{Optimal ET sequence|legend=1| 4, 15f, 19, 53def, 72def }}
Line 126: Line 121:
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
* 13-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 320.000]
* 15-odd-limit diamond monotone and tradeoff: ~6/5 = 315.789


{{Optimal ET sequence|legend=1| 15, 19, 34, 87ddee }}
{{Optimal ET sequence|legend=1| 15, 19, 34, 87ddee }}
Line 212: Line 205:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [317.647, 319.265]


{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
Line 230: Line 222:
* 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
* 11-odd-limit diamond monotone and tradeoff: ~6/5 = [317.647, 320.000]


{{Optimal ET sequence|legend=1| 15, 34d, 49, 181bbcdddeee }}
{{Optimal ET sequence|legend=1| 15, 34d, 49, 181bbcdddeee }}
Line 266: Line 257:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 317.263]


{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
Line 273: Line 263:


==== 2.3.5.7.13 subgroup ====
==== 2.3.5.7.13 subgroup ====
The S-expression-based comma list of this temperament is {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]](, [[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.
Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7.13


Comma list: 169/168, 225/224, 325/324
Comma list: 169/168, 225/224, 325/324
S-expression-based comma list: {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]](, [[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}


Sval mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
Sval mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
Line 299: Line 289:
* 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]
* 11-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 316.981]


{{Optimal ET sequence|legend=1| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}
{{Optimal ET sequence|legend=1| 19, 34de, 53, 72, 197e, 269ce, 341ce, 610bccee }}
Line 317: Line 306:
* 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~6/5 = [315.789, 316.981]


{{Optimal ET sequence|legend=1| 19, 34de, 53, 72, 125f, 197ef, 269cef }}
{{Optimal ET sequence|legend=1| 19, 34de, 53, 72, 125f, 197ef, 269cef }}
Line 469: Line 457:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [316.667, 317.647] (19\72 to 9\34)
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [316.667, 317.647] (19\72 to 9\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~6/5 = [316.667, 317.263]


{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473, 806b }}
Line 487: Line 474:
* 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
* 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]
* 11-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.370]


{{Optimal ET sequence|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
{{Optimal ET sequence|legend=1| 34, 53, 87, 140, 227, 367e, 507e }}
Line 506: Line 492:
* 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
* 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
* 13-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.647]
* 15-odd-limit diamond monotone and tradeoff: ~6/5 = [316.981, 317.241]


{{Optimal ET sequence|legend=1| 34, 53, 87, 140, 367e, 507e }}
{{Optimal ET sequence|legend=1| 34, 53, 87, 140, 367e, 507e }}
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